## Impact of vertical stratification of inherent optical properties on radiative transfer in a plane-parallel turbid medium

Optics Express, Vol. 18, Issue 6, pp. 5629-5638 (2010)

http://dx.doi.org/10.1364/OE.18.005629

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### Abstract

The atmosphere is often divided into several homogeneous layers in simulations of radiative transfer in plane-parallel media. This artificial stratification introduces discontinuities in the vertical distribution of the inherent optical properties at boundaries between layers, which result in discontinuous radiances and irradiances at layer interfaces, which lead to errors in the radiative transfer simulations. To investigate the effect of the vertical discontinuity of the atmosphere on radiative transfer simulations, a simple two layer model with only aerosols and molecules and no gas absorption is used. The results show that errors larger than 10% for radiances and several percent for irradiances could be introduced if the atmosphere is not layered properly.

© 2010 OSA

## 1. Introduction

1. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete ordinate method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. **27**(12), 2502–2509 (1988). [CrossRef] [PubMed]

## 2. Derivation of the discontinuity and its underlying physics

1. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete ordinate method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. **27**(12), 2502–2509 (1988). [CrossRef] [PubMed]

*μ*is the cosine of zenith angle (Fig. 1 ), positive for downward and negative for upward directions,

*ϕ*is azimuth angle,

*τ*is the optical depth,

**is intensity (radiance) of the radiation field, and**

*I***is source function given by:**

*J**ω*is the single scattering albedo,

*P*is the scattering phase function due to single scattering,

*πF*is the solar irradiance at the top of the atmosphere (TOA), and

_{0}*μ*and

_{0}*ϕ*are the cosine of the solar zenith angle and solar azimuth angle, respectively. The optical depth,

_{0}*τ,*is given through integration of extinction coefficient

*β*:

*b*is the total optical thickness of the atmosphere, the symbols ↑ and ↓ denote the upward and downward radiance, respectively. In the Eqs. (4) and (5), the values of

*μ*are always positive, that is

*μ*= |

*μ*|>0.

^{↑}and I

^{↓}for the horizontal direction (

*μ*= 0) at τ should have exactly the same value if the atmospheric optical parameters vary continuously with optical depth. Therefore, the following equality must be satisfied:

*ω*and phase function

*P,*differ between neighboring layers. Thus, the IOPs are discontinuities across layer boundaries. In this situation, Eq. (6) may not be satisfied. To provide direct insight into the consequence of this artificial discontinuity, a two-layer model (Fig. 1) is used to simulate the radiances and irradiances. The single scattering albedo and scattering phase function are set to be

*ω*, and

_{x}*P*for layer

_{x}*x*, where

*x =*1, 2 (for details, see section 3).

### 2.1 Single Scattering Approximation

*μ*= 0, the single scattering radiance in the horizontal direction becomes:

### 2.2 Multiple Scattering Radiance

*J*in Eqs. (4) and (5) is replaced by:where the subscript

*“n”*stands for the n

^{th}order of scattering. Inserting Eq. (13) into Eq. (4) and letting

*μ*tend to 0, we find that the first term of Eq. (4) tends to zero, and the second term tends to

*n*

^{th}order scattering resulting from the expression for the downward radiance (see Fig. 1) can be written as:and, similarly, the horizontal radiance resulting from the expression for the upward radiance becomes:

*J*by the total source function

_{n}*J*in Eqs. (14) and (15). Comparing Eq. (16) with (12), we easily see that Eq. (12) is a special case of Eq. (16) for the single scattering case for which n = 1.

## 3. Case study

*a*and

*m*, respectively, stand for aerosol and molecule, and “0” means the coefficients at height

*z*= 0km. The column integrated optical depths due to aerosol extinction and molecular (Rayleigh) scattering are

*τ*= 2

_{a}*β*

_{a}_{,}

*,*

_{0}*τ*= 8

_{m}*β*

_{m}_{,}

*.*

_{0}8. J. Xie and X. Xia, “Long-term trend in aerosol optical depth from 1980 to 2001 in north China,” Particuology **6**(2), 106–111 (2008). [CrossRef]

9. P. M. Teillet, “Rayleigh optical depth comparisons from various sources,” Appl. Opt. **29**(13), 1897–1900 (1990). [CrossRef] [PubMed]

*λ*is the wavelength in micrometers (

*μ*m). We use a total Rayleigh scattering optical depth of 0.316, which corresponds to

*λ*= 0.415

*μ*m, which is one of the wavelengths used in the Multi-Filter Rotating Shadowband Radiometer (MFRSR) designed for retrieval of aerosol and cloud optical depths [10

10. L. Harrison and J. Michalsky, “Objective algorithms for the retrieval of optical depths from ground-based measurements,” Appl. Opt. **33**(22), 5126–5132 (1994). [CrossRef] [PubMed]

15. M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. **59**(3), 524–543 (2002). [CrossRef]

*μ*m, which is adopted in instruments used in ocean color satellite remote sensing [16

16. H. R. Gordon and D. J. Castaño, “Coastal zone color scanner atmospheric effects and its application algorithm: multiple scattering effects,” Appl. Opt. **26**(11), 2111–2122 (1987). [CrossRef] [PubMed]

17. M. Wang and H. R. Gordon, “Retrieval of the columnar aerosol phase function and single-scatting albedo from sky radiance over the ocean: simulations,” Appl. Opt. **32**(24), 4598–4609 (1993). [CrossRef] [PubMed]

*P*(Fig. 2 ) for randomly-oriented prolate spheroids with aspect ratio a/b = 4.0, size parameter x = 10.079368, and a refractive index m = 1.55-0.01

_{a}*i*is used [18

18. F. Kuik, J. F. De Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. **47**(6), 477–489 (1992). [CrossRef]

*P*and the aerosol single scattering albedo

_{a}*ω*are assumed to be constant in each layer. As already mentioned, we further assume that there is no gas absorption. If the whole atmosphere is separated into several layers, and each layer is assumed to be homogeneous, then the optical depth

_{α}*Δτ*, single scattering albedo

_{x}*ω*, and phase function

_{x}*P*for a layer

_{x}*x*located between

*z*

_{1}and

*z*

_{2}can be written as: where

*τ*and

_{a,x}*τ*, respectively, are the contributions of aerosol and Rayleigh scattering.

_{m,x}1. K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete ordinate method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. **27**(12), 2502–2509 (1988). [CrossRef] [PubMed]

*z*= 2 km, as shown in Fig. 1. Thus, the aerosol optical depth of layers 1 and 2 are 0.147 and 0.253 respectively, and the Rayleigh scattering optical depth of layers 1 and 2 are 0.246 and 0.07, respectively. The solar zenith angle is set to be 53.13 degrees (

*μ*

_{0}= 0.6). All the results shown in this paper are normalized by multiplying by 1/

*F*.

_{0}### 3.1 Horizontal radiance at layer boundary

*z =*2 km) between the two layers are shown in Fig. 3 as a function of the aerosol single scattering albedo

*ω*, and the errors of the horizontal radiances given by the 2-layer model are also illustrated. “2km+” and “2km-” denote the horizontal radiances (

_{a}*μ*= 0), which are calculated from the downward radiance in the upper layer (“2km+”), and from the upward radiance in the lower layer (“2km-”) of the 2-layer model. The “true” value is calculated from the improved successive order of scattering method SOSVRT [5

5. Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. **87**(3-4), 243–259 (2004). [CrossRef]

*μ*= 0 produced by the 2-layer model? The reason is that the atmosphere is broken into two different layers, but each layer is assumed to be homogenous. Thus, this 2-layer model is a very crude representation of the true vertical variation of the atmosphere. As a result, the atmospheric IOPs on one side of the interface between the 2 layers are different from the IOPs on the other side. This artificial discontinuity in atmospheric IOPs results in different horizontal radiances. Figure 3 shows that the differences increase with the increase in aerosol absorption (smaller single scattering albedo). Because we assume there is no molecular absorption in atmosphere, the smaller

*ω*results in bigger difference of the

_{α}*ω*(or

*ωP*) between the two layers and larger errors in the radiances.

19. M. Duan, Q. Min, and J. Li, “A fast radiative transfer model for simulating high-resolution absorption bands,” J. Geophys. Res. **110**(D15), D15201 (2005), doi:. [CrossRef]

*ω*and

_{x}*P*for the two layers, resulting in the difference in radiances. For example, if the lower layer contains absorbing aerosols (

_{x}*ω*= 0.5), whereas the upper layer contains molecules only, the single scattering albedo just below the interface of the two layers is 0.5, while it is 1.0 just above the interface.

_{a}### 3.2 Effects on calculation of radiances and irradiances

*ω, Δτ*and

*P*for each layer are given by Eqs. (20), (21). The “true” value is calculated from the improved version of SOSVRT by direct integration of the source function over

*τ*with a small step size

*δτ*; in the case study of this paper, we used

*δτ*≈0.002. For each integration step between

*τ*and

_{i}*τ*+

_{i}*δτ*, the extinction coefficients

*β*and

_{a}*β*at three points,

_{m}*τ*,

_{i}*τ*+

_{i}*δτ*/2 and

*τ*+

_{i}*δτ*are computed through Eqs. (3), (17) and (18), and the single scattering albedos

*ω*and phase functions

*P*at the three points are given through Eqs. (21) and (22) by replacing optical depth

*Δτ*with extinction coefficient

*β*. Allowing for this vertical variation within each layer in the SOSVRT is the main difference from a traditional n-layer model in which each layer is assumed to be homogeneous. Then the source functions can be calculated with Eq. (7) or (13) and integrated to compute the radiance by assuming it varies linear-exponentially with

*τ*[5

5. Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. **87**(3-4), 243–259 (2004). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete ordinate method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. |

2. | A. Berk, L. S. Bernstein, and D. C. Robertson, “MODTRAN: A moderate resolution model for LOWTRAN 7,” GL-TR-89–0122, Phillips Laboratory, ADA214337 (1989). |

3. | K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. |

4. | E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. |

5. | Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. |

6. | M. Duan and Q. Min, “A polarized radiative transfer model based on successive order of scattering method,” Adv. Atmos. Sci. Doi: (in print). |

7. | G. E. Thomas, and K. Stamnes, |

8. | J. Xie and X. Xia, “Long-term trend in aerosol optical depth from 1980 to 2001 in north China,” Particuology |

9. | P. M. Teillet, “Rayleigh optical depth comparisons from various sources,” Appl. Opt. |

10. | L. Harrison and J. Michalsky, “Objective algorithms for the retrieval of optical depths from ground-based measurements,” Appl. Opt. |

11. | L. Harrison, J. Michalsky, and J. Berndt, “Automated multifilter rotating shadow-band radiometer: an instrument for optical depth and radiation measurements,” Appl. Opt. |

12. | Q. Min and L. Harrison, “Cloud Properties Derived From Surface MFRSR Measurements and Comparison With GOES Results at the ARM SGP Site,” Geophys. Res. Lett. |

13. | Q. Min, E. Joseph, and M. Duan, “Retrievals of thin cloud optical depth from a multifilter rotating shadowband radiometer,” J. Geophys. Res. |

14. | M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. |

15. | M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. |

16. | H. R. Gordon and D. J. Castaño, “Coastal zone color scanner atmospheric effects and its application algorithm: multiple scattering effects,” Appl. Opt. |

17. | M. Wang and H. R. Gordon, “Retrieval of the columnar aerosol phase function and single-scatting albedo from sky radiance over the ocean: simulations,” Appl. Opt. |

18. | F. Kuik, J. F. De Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. |

19. | M. Duan, Q. Min, and J. Li, “A fast radiative transfer model for simulating high-resolution absorption bands,” J. Geophys. Res. |

**OCIS Codes**

(010.1310) Atmospheric and oceanic optics : Atmospheric scattering

(010.5620) Atmospheric and oceanic optics : Radiative transfer

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: December 14, 2009

Revised Manuscript: February 19, 2010

Manuscript Accepted: March 1, 2010

Published: March 4, 2010

**Citation**

Minzheng Duan, Qilong Min, and Knut Stamnes, "Impact of vertical stratification of inherent optical properties on radiative transfer in a plane-parallel turbid medium," Opt. Express **18**, 5629-5638 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-5629

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### References

- K. Stamnes, S. C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete ordinate method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27(12), 2502–2509 (1988). [CrossRef] [PubMed]
- A. Berk, L. S. Bernstein, and D. C. Robertson, “MODTRAN: A moderate resolution model for LOWTRAN 7,” GL-TR-89–0122, Phillips Laboratory, ADA214337 (1989).
- K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991). [CrossRef]
- E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, and J.-J. Morcette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Rem. Sens. 35(3), 675–686 (1997). [CrossRef]
- Q. Min and M. Duan, “A successive order of scattering model for solving vector radiative transfer in the atmosphere,” J. Quant. Spectrosc. Radiat. Transf. 87(3-4), 243–259 (2004). [CrossRef]
- M. Duan and Q. Min, “A polarized radiative transfer model based on successive order of scattering method,” Adv. Atmos. Sci. Doi: (in print).
- G. E. Thomas, and K. Stamnes, Radiative transfer in the atmosphere and ocean. (Cambridge, 1999), P160.
- J. Xie and X. Xia, “Long-term trend in aerosol optical depth from 1980 to 2001 in north China,” Particuology 6(2), 106–111 (2008). [CrossRef]
- P. M. Teillet, “Rayleigh optical depth comparisons from various sources,” Appl. Opt. 29(13), 1897–1900 (1990). [CrossRef] [PubMed]
- L. Harrison and J. Michalsky, “Objective algorithms for the retrieval of optical depths from ground-based measurements,” Appl. Opt. 33(22), 5126–5132 (1994). [CrossRef] [PubMed]
- L. Harrison, J. Michalsky, and J. Berndt, “Automated multifilter rotating shadow-band radiometer: an instrument for optical depth and radiation measurements,” Appl. Opt. 33(22), 5118–5125 (1994). [CrossRef] [PubMed]
- Q. Min and L. Harrison, “Cloud Properties Derived From Surface MFRSR Measurements and Comparison With GOES Results at the ARM SGP Site,” Geophys. Res. Lett. 23(13), 1641 (1996). [CrossRef]
- Q. Min, E. Joseph, and M. Duan, “Retrievals of thin cloud optical depth from a multifilter rotating shadowband radiometer,” J. Geophys. Res. 109(D2), D02201 (2004), doi:. [CrossRef]
- M. Alexandrov, A. Marshak, B. Cairns, A. A. Lacis, and B. E. Carlson, “Automated cloud screening algorithm for MFRSR data,” Geophys. Res. Lett. 31(4), L04118 (2004), doi:. [CrossRef]
- M. Alexandrov, A. Lacis, B. Carlson, and B. Cairns, “Remote sensing of atmospheric aerosols and trace gases by means of multifilter rotating shadowband radiometer. Part I: Retrieval algorithm,” J. Atmos. Sci. 59(3), 524–543 (2002). [CrossRef]
- H. R. Gordon and D. J. Castaño, “Coastal zone color scanner atmospheric effects and its application algorithm: multiple scattering effects,” Appl. Opt. 26(11), 2111–2122 (1987). [CrossRef] [PubMed]
- M. Wang and H. R. Gordon, “Retrieval of the columnar aerosol phase function and single-scatting albedo from sky radiance over the ocean: simulations,” Appl. Opt. 32(24), 4598–4609 (1993). [CrossRef] [PubMed]
- F. Kuik, J. F. De Haan, and J. W. Hovenier, “Benchmark results for single scattering by spheroids,” J. Quant. Spectrosc. Radiat. Transf. 47(6), 477–489 (1992). [CrossRef]
- M. Duan, Q. Min, and J. Li, “A fast radiative transfer model for simulating high-resolution absorption bands,” J. Geophys. Res. 110(D15), D15201 (2005), doi:. [CrossRef]

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